How Many Coincidences Are Enough?

Today the phone rang, and, rather oddly, the phone that happened to be right next to me was dead.  So I took it back to its base in my parents’ room (the fact that my parents’ phone was out of their room and the one right next to me was also unusual.) While I was there, the phone rang again.  Even though it seemed to be working fine on the charger, I answered using the fax machine just in case it would have died during the phone conversation. The call was from my dad.  He wanted me to go to his room and tell him the model number of his fax machine. An unusual request. Obviously, since I was standing in front of it, I didn’t have to so much as turn my head. I use the fax machine’s phone probably much less than once a year. 

Not that it’s a stupendous coincidence, but the point is this: how would we calculate the odds of this coincidence?

We could take the odds that I would have bothered to answer at all (a), and multiply by the odds that the nearest phone would have been my parents’ (b), and multiply that by the odds that it would have been dead (c), and assume that I would have put it back on its base and assume that my dad would have tried calling again and multiply that by the odds that I would have answered with the fax machine instead of the better phone that seemed to have been working fine (d). How would we determine a, b and c and d? Perhaps we don’t know how often I answer. Or perhaps we do, but I answer more often at different times of the day. Do we consider what time of day my dad called? If so, do we also consider the likelihood that he would have called at that time, or is his calling at that time a given? Do we consider what I was doing at the time? How would we possibly quantify that? Or do we could just replace all that with what percentage of phone calls I answer with the fax machine?

So now we have the other part of the coincidence: the probability of my dad wanting to see the model number of the fax machine. Do we start with the probability that it would be my dad? Or is it a given that it’s my dad because if he didn’t call there would have been no phone call? Which is primary? Getting a phone call, or my dad wanting to know the model number? How do we even approach computing the probability that he would want to know the model number?

Or is the coincidence just about me not having to move from that spot or turn my head to do something I needed to at that moment? What if I only had to turn my head a little, and/or step a little bit in some direction? According to what functions would we translate that into a probability? And then we have to compute the probabilities of all the things that could happen in which I’d need to…read some piece of information? Or should it be more general, including, e.g., pushing a button? But in that case, what if it’s just a conversation? That doesn’t require turning my head at all­! That notwithstanding, do we allow for the coincidence to be anywhere in the house? What if I had happened to answer it in the family room and someone needed to know something about that thing in the family room? And the probability that I would have been in the family room is a completely different value…

Or is the essential thing to compute not the improbability that what I needed was right there, but the improbability of the convenience itself more generally? Then we’d have to know the probabilities of every possible thing that could happen of the same or lower general convenience level, in order to calculate the improbability of one of them happening…one of them happening in what period of time? That day? That year? My life? Everyone’s life? If you were looking for a convenience in my life for that day, then you’d need the probability that such a thing would happen to me that day. But what time period are we really looking at? Am I amazed because it happened at that moment? Or because it happened out of my entire life? Or something in between? (The coincidence cited above isn’t quite amazing, but it’s just a random example. The same general flavor of ambiguity applies to most kinds of coincidences.) We’d have to compare the numbers of the occurrences of every type of convenience that happened during some undefined time period to the frequencies at which they would happen by chance.

Or if I am conveying the story to another, to know the amazingness-level (that’s what we’re really trying to measure here) of the coincidence happening, he has to compare it to the likelihood of it happening, or to anything of similar likelihood happening, during some undefined time period to…anyone he knows? Anyone who would have told him about it? Anyone he knows multiplied by the probability of the given person telling him about it? What does he do if someone tells him about it happening to someone else? What if he’s reading about the occurrence on the Internet? Does he consider how much he reads, what things he’s looks for, what types of things people usually put on the internet, what he was looking for at that moment, how many people use the internet, etc.? So, what would he consider, and how would he formalize/quantify those things, let alone find out their values?

And let’s not forget that convenience is just one type of possible coincidence. Let’s say that there are X different types of possible coincidences that could have happened to me that day. Then the wonderment of a coincidence of convenience happening seems somewhat minified, so one would think that should be considered. But then we need to know X, and the likelihoods of coincidences of the same caliber of every possible type. But isn’t the categorization system itself arbitrary, thus calling for a continuum of possible events in a space of an undefined number of dimensions to compare this event to? Then how would we calculate the likelihood of this coincidence? More to the point, the question devours itself, since we’re left trying to calculate the unlikelihood of an unlikelihood happening (remember, this is about amazement per se), and statistics requires parameters. It can only determine the unlikelihood of something within a greater uniformity.

Now enter the skeptics’ assumption that synchronicity only seems to be a real thing because we forget all the times something coincidental doesn’t happen and remember the times it does. To know if that’s true, one would have to know whether the degrees and frequencies at which apparent synchronicities actually occur are objectively much higher than would be determined by pure chance. But as I have just shown, it is completely impractical, if not technically impossible, to calculate those things. Even if one could, one wouldn’t because it would be too difficult and involved. So, that particular proffering of disenchantment by the skeptics can at best be a theory, but skeptics take it to be the truth. It is therefore an assumption, reflecting only their own biases. It makes one wonder what other things self-proclaimed “skeptics” baselessly assume…

I wrote more on this subject here:

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